Global Fractional Analytic Perturbation Theory in QCD with Selected Applications

We give the generalization of Fractional Analytic Perturbation Theory (FAPT) for QCD observables, recently developed both for the Euclidean and Minkowski regions of the squared momentum transfer q(2), which takes into account heavy-quark thresholds. The original analytic approach to QCD, initiated by Jones, Solovtsov, and Shirkov, is briefly summarized. We also shortly consider the basic aspects of FAPT, and then, concentrate on the accounting for the problem of heavy-quark thresholds and the construction of a global version of FAPT. We discuss what one should use as an analytic coupling in the timelike region q(2) = s > 0 for the e(+)e(-) annihilation and the pion's form factor, and consider applications to phenomenologically relevant processes (the factorizable part of the pion's form factor and the Higgs boson decay into a b (b) over bar pair), as well as to the summation of perturbative series.